Vale will go through to the next round and Page said: "I've said to them in the changing room after they've got to take the confidence and believe in themselves for what they can do against a good team in Burnley.
"They've got to take that now into the league again. There were a lot of positives on Saturday against Crewe, we took that into tonight against an excellent team. We've just got to keep building this momentum."
Having seen his side beaten, Sean Dyche said: "It's the madness of losing a game. Sometimes you have to look at what you did and not just the result.
"I thought it was a good performance from Port Vale and ours was good with the ball, but we didn't really have the bite that we normally have in the top third.
"I thought defensively we were excellent and particularly our two centre halves, who were outstanding and limited them to very little. Unfortunately for Matt Gilks he's had to pull the ball out of the net. He's hardly had a save to make because of the back four in front of him.
"In the end they got a goal that was over the line, but how the referee gave it when his linesman didn't is a physical impossibility, but that's by the by. In the end we didn't create the chances until right at the death after they scored, but you have to react when it's 0-0, not when you are 1-0 down."
Having made four team changes from the team that started at Leeds, he added: "We've made changes because we have to make sure people are fit and sharp, but we have to create better chances. We definitely have to add that to our game.
"But you have to play games to get that because you can't just flick a switch. We felt this was a game that needed changes, but I must make it clear if you look at our team there was enough quality and experience in there, so we had enough to win that game.
"The other thing that has changed, and the players know this because I've told them all pre-season, is that we are now a scalp. Two years ago things were different, but now we've just been in the Premier League so the mentality has to change.
"Take nothing away from Port Vale though because they played hard and tried to upset us and got men behind the ball. All of those things, but we had enough to win the game and we didn't do our job."